156 Logic as a Tool It is important to note that the validity of some of these forms depends on an assumption which Aristotle made about the interpretation of the quantifiers in the types A and E, known as Existential Import: in the propositions “All S are P” and “No S are P” the predicate S is non-empty, that is, there are S objects Under this assumption, these universal proposition types, which are at the top of the Square of Opposition, logically imply respectively their existential counterparts at the bottom, “Some S are P” and “Some S are not P.” The Existential Import assumption is compliant with the normal natural language usage of such expressions, but is not formally justified by the semantics of first-order logic that we have assumed; we must therefore handle it with special care and will only apply it in the context of syllogisms Without this assumption, nine of the syllogistic forms listed above cease to be valid I leave it as an exercise for the reader to identify these Note that the syllogistic forms are not formulae but inference rules It is however possible to develop a deductive system for deriving some valid syllogisms from others; see the references for further details To summarize, categorical syllogisms capture a natural and important fragment of logical reasoning in first-order logic involving only unary predicates and constant symbols They not capture simple sentences involving binary predicates, such as “Every man loves a woman”, but have one great technical advantage: the simple decision procedure for testing valid syllogisms given by the method of Venn diagrams As I explain in the next chapter, no such decision procedure exists for first-order logic in general, even when the language involves a single binary predicate References for further reading For more details on syllogisms see Carroll (1897), Barwise and Echemendy (1999), and van Benthem et al (2014), plus many books on philosophical logic Exercises 3.5.1 For each of the syllogisms listed in Example 121, verify its type and figure as claimed in the text 3.5.2 For each of the following syllogisms (taken from Section 3.4.8, Exercise 17(a–f)) identify its major, minor, and middle terms and its type and figure (c) (a) All philosophers are humans All humans are mortal All philosophers are mortal Some negative numbers are rationals All integers are rationals Some integers are negative numbers (d) (b) No work is fun Some entertainment is work Some entertainment is not fun All bank managers are rich No bank managers are teachers No teachers are rich